On the structure and zero divisors of the Cayley-Dickson sedenion algebra

The algebras ℂ (complex numbers), ℍ (quaternions), and (octonions) are real division algebras obtained from the real numbers ℝ by a doubling procedure called the Cayley-Dickson Process. By doubling ℝ (dim 1), we obtain ℂ (dim 2), then ℂ produces ℍ (dim 4), and ℍ yields (dim 8). The next doubling process applied to then yields an algebra (dim 16) called the sedenions. This study deals with the subalgebra structure of the sedenion algebra and its zero divisors. In particular, it shows that has subalgebras isomorphic to ℝ, ℂ, ℍ, , and a newly identified algebra ̃ called the quasi-octonions that contains the zero-divisors of .